A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

Aili Zhang,Zhang Liu

doi:10.1155/2020/6282869

Abstract

This paper focuses on the De Finetti’s dividend problem for the spectrally negative Lévy risk process, where the dividend is deducted from the surplus process according to the racheting dividend strategy which was firstly introduced in Albrecher et al. (2018). A major feature of the racheting strategy lies in which the dividend rate never decreases. Unlike the conventional studies, the closed form expression for the expected, accumulated, and discounted dividend payments until the draw-down time (rather than the ruin time) is obtained in terms of the scale functions corresponding to the underlying Lévy process. The optimal barrier for the ratcheting strategy is also studied, where the dividend rate can be increased. Finally, two special cases, where the scale functions are explicitly known, i.e., the Brownian motion with drift and the compound Poisson model, are considered to illustrate the main result.

Highlights

During the first half of the 20th century, the actuaries concentrated on assessing the stability of an insurance company via the probability of ruin
In [1], where the surplus dynamics followed a discrete time random walk, De Finetti showed that a barrier dividend strategy is the optimal dividend strategy because it produces the maximum firm value of the company
The optimality of the barrier dividend strategy has been proved for various risk models under suitable assumptions, see, for instance, Loeffen [3], Loeffen and Renaud [4], Yin and Wang [5], Yuen and Yin [6], Wang and Zhou [7], Yu et al [8], and the references therein

Summary

Introduction

During the first half of the 20th century, the actuaries concentrated on assessing the stability of an insurance company via the probability of ruin. In practice, the practitioners are unwilling to accept a reduction in the rate of the dividend payment stream because a decrease in the rate of paying dividends may lead to negative psychological impacts on shareholders and the firm value of the company Taking into account these considerations, Albrecher et al [16] introduced a dividend strategy called ratcheting strategy, where the dividend rate would never decrease over time, but would increase once the underlying process hits some b and stay at this higher level until the time of ruin. Wang and Zhou [20] introduced the concept of draw-down Parisian ruin time for a spectrally negative Levy risk process and obtained the kth moment of the discounted total dividends paid according to the barrier dividend strategy until the draw-down Parisian ruin time, which generalized a result of Czarna and Palmowski [21].

Problem Formulation

Problem Presentation and the Main Results

Findings

Special Cases